50 111 METRIC SPACES

(3.13.5) Suppose a' = lim f(x). Then, for every subset B c A such that

x-+a, jceA

fl e B, o! is also the limit off at the point a, with respect to B. This applies
in particular when B = V n A, where V is a neighborhood of a.

Obvious consequence of the definition and (3.11.6).

(3.13.6) Suppose f has a limit a' at the point a e A with respect to A; if g is
a mapping ofE' into E", continuous at the point a'y then g(a') = lim g(f(x)).

x-*a, x e A

This follows at once from (3.11.5).

(3.13.7) 7/a' = lim /(*), then a' e/(A).

a, xeA

For by (3.13.1), for every neighborhood V of a', V n/(A) contains
n A), which is not empty since a e A.

An important case is that of limits of sequences: in the extended real line,
\ve consider the point +00, which is a cluster point of the set N of natural
integers. A mapping of N into a metric space E is a sequence n -> xn of points
of E; if a e E is limit of that mapping at + oo, with respect to N, we say that
a is limit of the sequence (x^) (or that the sequence (xn) converges to a) and
write a = lim xn. The criteria (3.13.1) and (3.13.2) become here:

(3.13.8) In order that a = lim jcn, a necessary and sufficient condition is that,

n-* oo

for every neighborhood V of a, there exist an integer nQ such that the relation
n^nQ implies xn e V (in other words, V contains all xn with the exception
of a finite number of indices).

(3.13.9) In order that a= lim jcn, a necessary and sufficient condition is

w~+ao

that, for every e > 0, there exist an integer nQ such that the relation n^nQ
implies d(a, xn) < e,

This last criterion can also be written lim d(a, xn) = 0.

H-*.00

A subsequence of an infinite sequence (%n) is a sequence k -> xnk, where
k -> nk is a strictly increasing infinite sequence of integers. It follows at once
from (3.13.5) that:ous mapping of the space QJ. of rational numbers > 0